A particularly efficient implementation of offset continuation results from a log-stretch transform of the time coordinate Bolondi et al. (1982), followed by a Fourier transform of the stretched time axis. After these transforms, equation () from Chapter takes the form
Analogously to the case of the plane-wave-destructor filters discussed in the previous section, we can construct an effective offset-continuation finite-difference filter by studying first the problem of wave extrapolation between neighboring offsets. In the frequency-wavenumber domain, the extrapolation operator is defined in accordance with equation (), as follows:
Returning to the original domain, I approximate the continuation operator with a finite-difference filter of the form
Figure compares the phase characteristic of the finite-difference extrapolators () with the phase characteristics of the exact operator () and the asymptotic operator (). The match between different phases is poor for very low frequencies (left plot in Figure ) but sufficiently accurate for frequencies in the typical bandwidth of seismic data (right plot in Figure ).
Figure compares impulse responses of the inverse DMO operator constructed by the asymptotic operator with those constructed by finite-difference offset continuation. Neglecting subtle phase inaccuracies at large dips, the two images look similar, which indicates the high accuracy of the proposed finite-difference scheme.
When applied on the offset-midpoint plane of an individual frequency slice, the one-dimensional implicit filter () transforms to a two-dimensional explicit filter with the 2-D Z-transform